A general formula for Hecke-type false theta functions

In recent work where Matsusaka generalized the relationship between Habiro-type series and false theta functions after Hikami, he decomposed five families of Hecke-type double-sums of the form
\begin{equation*} \left( \sum_{r,s\ge 0 }-\sum_{r,s<0}\right)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}}, \end{equation*}
where $a,b,$ and $c$ are positive integers with negative discriminant $D:=b^2-ac<0$, into sums of products of theta functions and false theta functions. Here we obtain a general formula for such double-sums in terms of theta functions and false theta functions, which subsumes the decompositions of Matsusaka. Our general formula is similar in structure to the case of $D>0$, where Mortenson and Zwegers obtain a decomposition in terms of Appell functions and theta functions.